Dynamic characteristic difference of steel-spring floating slab track between single-carriage and multi-carriage models

Abstract

The steel-spring floating slab track (SSFST) is a low-stiffness structure, sensitive to the vehicle loads. Due to the coupling effect of the superposition of adjacent bogies, it is difficult for conventional single-carriage models to meet the simulation requirements. To find a balance between computation efficiency and authenticity of analytical model results, the influence of carriage number on SSFST should be studied. Based on the finite element method and multi-body dynamics, a refined three-dimensional coupled model of multi-carriage-SSFST-tunnel was established. The difference in the dynamic response of the SSFST between single-carriage and multi-carriage models was analyzed and compared with the measured data. The field test results show that structural displacements and accelerations under the two-carriage model are much closer to the measured data. The dynamic model analysis results show that the maximum displacement of the rail and SSFST in the midspan of the slab increase by 0.48 mm and 0.34 mm under the multi-carriage model, and the vibration reduction effectiveness increases by 1.4–2.0 dB. Dynamic responses of the rail and SSFST show minor differences under the two-carriage and three-carriage models. The article is expected to provide a reference for the theoretical research, design, and layout optimization of subway SSFST.

Keywords

Steel-spring floating slab track multi-carriage model dynamic characteristics subway vibration reduction

Introduction

With the rapid development of cities, the subway has constituted an integral part of modern urban transit. To reduce the impact of vibration on the surrounding environment, measures should be taken to reduce the vibration transmitted from underground railways into nearby buildings, especially the areas with high sensitivity to vibration and noise, such as hospitals, schools, and residential areas. At present, among all the vibration damping measures, the steel-spring floating slab track (SSFST) is the most effective and widely used.

The SSFST consists of five parts: lower foundation, steel-spring vibration isolator, concrete floating slab, track structure, and shear hinge. Figure 1 is a structure schematic diagram of SSFST. The basic vibration isolation principle of SSFST is to insert a linear resonator whose natural vibration frequency is much lower than the excitation frequency between the upper track structure and lower foundation. The concrete slab track with certain mass and stiffness is floated on the steel-spring isolator. The mass inertia of the floating slab is used to balance the dynamic load caused by moving trains, and only the unbalanced dynamic and static loads are transmitted to the subgrade or tunnel through the steel-spring isolators to achieve the purpose of vibration reduction.¹

Figure 1.

Structure diagram of steel-spring floating slab track.

At present, extensive research studies have been conducted on the prediction of vibration environmental response of the floating slab track. The analytical, semi-analytical, and numerical methods are most common used in vibration prediction. The analytical method is mainly used to analyze the dynamic response of moving load in half-space. There are a lot of simplifications and assumptions in the analytical model, which is more suitable for studying the law of vibration propagation. And it is difficult to achieve a quantitative prediction of vibration response. The semi-analytical method is to establish a complex analytical model and solve it by numerical methods. The pipe-in-pipe (PiP) model is a mature semi-analytical method applied in the vibration response of SSFST. The PiP model was developed by Forrest, Hunt, and Hussein,^2-4 and the vibration response can be solved in the frequency and wave number domain with high computational efficiency. With the development of computer technology, numerical methods are often used to solve vibration response problems of complex structures, including finite element method (FEM), boundary element method (BEM), and infinite element method (IFEM). Jee et al. analyzed the maximum length of the SSFST by the FEM according to the maximum additional stress criterion.⁵ Hou et al.⁶ established the coupled model of vehicle–track-surrounding buildings based on the FEM, and the vibration reduction effect of ballast track and SSFST from the perspective of time and frequency domain were studied. Lei et al. used the FEM to conduct modal analysis and harmonic response analysis of SSFST with different densities, thicknesses, and lengths.⁷ Li et al. compared the differences in vibration and structural characteristics of SSFST.⁸ Yang et al. established a coupled dynamic model of multi-carriage and SSFST and studied the influence of the stiffness enhancement measure on the dynamic performance of the system.⁹ The FEM takes too long to calculate a 3D model, while the BEM has developed rapidly in vibration response research because it can reduce integral dimension and greatly shorten calculation time. There are a lot of research on the combination of FEM and BEM, including the periodic FEM–BEM and wave number domain FEM and BEM. Gupta et al. calculated the structural dynamic response of SSFST by the FEM–BEM.^10,11 P. Galvin et al. established a multi-body FE–BE model and compared the vibration responses of the floating slab track under a high-speed moving load.¹² Lopes et al. studied the impact of SSFST vibration on the dynamic response of the surrounding buildings by 2.5D FE–PML (perfectly matched layers) method.¹³ Besides, the finite–infinite coupled model was conducted by combining the FEM and IFEM. Li et al. established a simplified FE–IFE coupled model and studied the influence of static preload on the vibration reduction effect of SSFST in the laboratory test.¹⁴ Yang et al. studied the dynamic response differences of the soil–tunnel system under moving train loads based on 2D and 2.5D FEM–IFEM.¹⁵

In the aforementioned numerical methods, in order to reduce the calculation time, the simulation of the train is mostly simplified into a single-carriage or moving loads. However, the single-carriage model cannot simulate the coupling effect of the load from the superposition of the rear bogie of the carriage in front and the front bogie of the carriage in rear. And the actual displacement and acceleration of the track system in the field are larger than the results obtained by the single-carriage model, which affects the judgment of whether the displacement exceeds the threshold and the vibration reduction effect reaches the standard. Numerical models established by the FEM have the advantages of solving complex, heterogeneous geometric, and nonlinear problems, more close to the on-site situations. Currently, for most vehicle–track–tunnel models, the tunnel is generally about 100 m long with hundreds of thousands of grid units. If the whole vehicle model is applied, the unit number of substructures will be too large, and the computational efficiency will be seriously decreased; meanwhile, if simulating the whole vehicle passing through, the calculation time will increase significantly. Therefore, it is necessary to research the difference of dynamic response of SSFST under single-carriage and multi-carriage models. Find a balance between a single-carriage model and a multi-carriage model: the computation time is reduced as much as possible, while the finite model is closer to the on-site situation.

To solve the aforementioned problems and compensate for the insufficient existing research, the innovations of this study can be summarized as follows.

The influence of multiple carriages on the dynamic response of the SSFST was studied from the perspective of statics and dynamics, and the field test was carried out to verify the difference.

A refined rigid-flexible coupled model of vehicle–SSFST–tunnel coupled finite element model was established, which fully takes into consideration the layout characteristics of the field SSFST and vibration isolators.

Coupling effect of adjacent carriages

The track structure is simplified as a continuous elastic foundation beam model (see Figure 2); the differential equation of the continuous elastic foundation beam model is shown in equation (1) under the single axle load p, where E is the elasticity modulus of rail; I_x is the section moment of inertia of rail; y is the rail deformation; K is the elasticity modulus of rail foundation; u is the stiffness under a rail; and b is the spacing of sleepers

E I_{x} \frac{d^{4} y}{d x^{4}} + K y = 0

(1)

K = \frac{u}{b}

(2)

Figure 2.

Continuous elastic foundation beam model under a single wheel load.

According to the boundary conditions listed below,

The rail angle is 0 at the loading point, which is ${(d y / d x)}_{x = 0} = 0$ ;

The rail shear force Q is equal to $P / 2$ at the loading point, which is ${(2 E I_{x} d^{3} y / d x^{3})}_{x = 0} = P$ ;

The rail displacement y is 0 at a wireless distance, which is x→∞, y = 0.

The rail deformation y and the rail stress F can be obtained

y = \frac{P}{8 E I_{x} β^{3}} e^{- β x} (\cos β x + \sin β x)

(3)

F = y^{} u = F_{\max}^{} e^{- β x} (\cos β x + \sin β x)

(4)

where

F_{\max} = P^{} \sqrt[4]{(b^{3} u / 64 E I_{x})}

and

β = \sqrt[4]{(K / 4 E I_{x})}

On the basis of $φ (β x) = e^{- β x} (\cos β x + \sin β x)$ , as shown in Figure 3,¹⁶ when βx ≥ 5, y ≈ 0 and F ≈ 0.

Figure 3.

Function relationship of βx and ψ(βx).

The stiffness of the common ballastless track is above 100 MPa; however, the stiffness of the SSFST is much lower than that of the common ballastless track.^17,18 Here, K is equal to 10–50 MPa, E is equal to 2.1 × 10⁵ MPa, and I_x is equal to 3.21 × 10⁻⁵ m⁴. The value x can be obtained and is equal to 4.28–6.41 m.

Because the rail deformation is symmetrical about the wheel–rail contact point, the foundation under the rail will bear different degrees of loads in the range of 4.28–6.41 m before and after the wheel–rail contact point under the single axle load P. For most subway vehicles, the distance between the front and rear bogies of adjacent carriages is about 5–7.5 m, and the axle distance is about 15–20 m (See Figure 4). Therefore, the coupling load effect on the rail foundation of the front and rear bogies on the same carriage is not coupled due to the long distance; however, there is a degree of coupling effect between the front and rear bogies of adjacent carriages on the rail foundation, as shown in Figure 5. Especially for the SSFST with low stiffness, this effect will be more prominent. It can be seen that when analyzing the dynamic response of the SSFST, the train load should not be solely treated as a single-carriage load, but the coupling effect of multiple carriages should be considered.

Figure 4.

Schematic diagram of subway vehicle structure.

Figure 5.

Coupling effect of loads from the front and rear bogies of adjacent carriages.

Simulation model and parameters

A rigid–flexible coupled model of vehicle–SSFST–tunnel is established by using the FEM, as shown in Figure 6. In this coupled model, the vehicle is considered as a multi-body system, and the rail, SSFST, and tunnel are considered as flexible bodies. The rigid and flexible models are coupled by nonlinear wheel–rail contact.

Figure 6.

Schematic of the coupling dynamic model.

Vehicle model

The vehicle model is established according to Type A subway vehicle, whose axle load is 17 tons. And its properties are shown in Table 1.¹⁹ The vehicle model consists of a car body, two bogies, and four wheelsets. The vehicle model has 31 DoFs (degrees of freedom). Concretely speaking, the car body and bogie have five DoFs in vertical, lateral, pitching, rolling, and yawing directions. Each wheelset owns four DoFs, whose longitudinal and pitching is not taken into consideration. The primary and secondary suspensions are regarded as spring-damper elements, and their stiffness and damping in longitudinal, vertical, and lateral directions are taken into consideration. The equation of the vehicle motion is as follows

M_{i} \{{\ddot{Z}}_{i}\} + C_{i} \{{\dot{Z}}_{i}\} + K_{i} \{Z_{i}\} = \{P_{i}\} (i = 1,2, \dots, N_{v})

(5)

where

N_{v}

is the number of the carriages;

{\ddot{Z}}_{i}

{\dot{Z}}_{i}

, and

Z_{i}

denote the acceleration, velocity, and displacement vector of the ith carriage, respectively; M _i, C _i, and K _i denote the mass, stiffness, and damping matrices of the ith carriage, respectively; and P_i denotes the external force of the ith carriage.

Table 1.

Properties of the vehicle.

Property	Magnitude	Unit
Car body mass	54,400	kg
Bogie mass	3600	kg
Wheelset mass	1700	kg
Axle distance	15.7	m
Bogie distance	2.5	m

Considering that the vehicle passes through the SSFST at a uniform speed, the longitudinal forces of the coupler buffer device are very small, and the lateral and vertical interactions between carriages can be ignored. Therefore, the connection between carriages is simplified as no connections.

Track model

The rails are modeled as beam elements. The rails are connected to the SSFST by fasteners, simulated by spring-damper elements, and the rotation of the end is restricted. The SSFST is modeled as solid elements, and the length of a single slab is 25 m. The steel-spring vibration isolators are set between SSFST and concrete slab, and the isolators are simulated by 3 × 3 three-direction spring-damper elements. The stiffness of vibration isolators in the midspan and end of the slab is 7.50 kN/mm and 5.33 kN/mm, respectively, and the isolators in the slab end are arranged tightly for reducing the displacement. The longitudinal and horizontal distance of isolators are 1.20 m and 1.96 m, respectively. And the layout of the isolators is shown in Figure 7. The shear hinge is arranged at the end of the SSFST to reduce the deformation discontinuity and enhance the integral stiffness of the track. The schematic of the SSFST model is shown in Figure 8, and the main properties are shown in Table 2.

Figure 7.

Layout of the steel-spring isolators.

Figure 8.

Schematic of the steel-spring floating slab track model.

Table 2.

Properties of SSFST.

Property		Magnitude	Property		Magnitude
Rail elasticity modulus (Pa)		2.1 × 10¹¹	Rail Poisson’s ratio		0.3
Fastener stiffness (N/m)	Vertical	3.5 × 10⁷	Fastener damping (N·s/m)	Vertical	6.0 × 10⁴
	Lateral	2.0 × 10⁷		Lateral
	Longitudinal	2.0 × 10⁷		Longitudinal
SSFST	Elasticity modulus (Pa)	3.25 × 10¹⁰	Shear hinge	Elasticity modulus (Pa)	2.06 × 10¹¹
	Poisson’s ratio	0.167		Poisson’s ratio	0.29
	Density (kg/m³)	2.5 × 10³		Density (kg/m³)	7.85 × 10³

Note: SSFST: steel-spring floating slab track.

Tunnel model

To eliminate the effects of boundary conditions, the tunnel is modeled as a 60 m × 100 m × 150 m soil structure, whose cross section is rectangular, according to the practical situation of the field, as shown in Figure 9. The concrete slab, tunnel segment, and tunnel soil are all modeled as solid elements. The soil is simulated by using the Mohr–Coulomb model, and the concrete slab and tunnel segment are simulated by using the linear elastic model. The properties are shown in Table 3. This study assumes the contact between the tunnel soil and tunnel segment and the tunnel segment and the concrete slab is in a good manner, and the relative displacement between structures is not considered.

Figure 9.

Finite element model of the tunnel: (a) rectangular tunnel; (b) tunnel model.

Table 3.

Properties of the tunnel.

Item	Constitutive model	Elasticity modulus (GPa)	Poisson’s ratio	Density (kg/m³)	Frictional angle (°C)	Cohesive force (GPa)
Tunnel soil	Mohr–coulomb	3.0	0.280	2448	45	1.2
Tunnel segment	Linear elastic	31.5	0.167	2500	–—	–—
Concrete slab	Linear elastic	32.5	0.167	2500	–—	–—

Wheel–rail contact model

The vertical contact force between wheel and rail is determined by using the Hertz contact model. The wheel–rail normal force p(t) is calculated according to the Hertzian nonlinear elastic contact theory,²⁰ as shown in equation (6), where G is the wheel–rail contact constant and δZ(t) is the elastic compression between wheel and rail. When considering displacement irregularity Z₀(t), the wheel–rail force equation is shown as equation (7), where Z_wj(t) is the displacement of the ith wheelset at time t and Z_r(x_wj,t) is the displacement of the rail under the ith wheelset at time t. The tangential force is determined by using the Coulomb friction model. The Coulomb friction model defines the tangential force τ as shown in equation (8), where μ is the coefficient of friction

p (t) = {[\frac{1}{G} δ Z (t)]}^{3 / 2}

(6)

P_{j} (t) = {\frac{1}{G} [Z_{w j} (t) - Z_{r} (x_{w j}, t) - Z_{0} (t)]}^{3 / 2}

(7)

τ = μ p (t)

(8)

Track irregularity and contact relation

The track irregularity is the measured irregularity power spectrum density (PSD) X(k), and the inverse Fourier method is adopted to transform the PSD.²¹ Then, the time-domain function of track irregularity x(n) is obtained, as shown in equation (9), where N_r is the sampling number. The generated sample of track irregularity is shown in Figure 10. The irregularity spatial-domain samples are applied to the left and right rails, respectively, by applying vertical and lateral displacement to the grid nodes of the rails

x (n) = \frac{1}{N_{r}} \sum_{k = 0}^{N_{r} - 1} X (k) e^{(i 2 π k n / N_{r})} (n = 0,1, \dots, N_{r} - 1)

(9)

Figure 10.

Track irregularity samples: (a) left rail; (b) right rail.

For the contact of the other structural components, assuming that the contact between the slab track, backfill layer, and the lining is in a good condition, the relative displacement between layers is not considered, and the tie constraint is used for the connection. Considering the actual situation of the structures, both ends of the rails, track structures, and tunnel are symmetric constraints.

Vibration equation of the whole system

The coupled equations of motion for the vehicle–SSFST–tunnel system can be derived as follows

[\begin{matrix} M_{v v} & 0 & 0 \\ 0 & M_{s s} & 0 \\ 0 & 0 & M_{t t} \end{matrix}] {\begin{matrix} {\ddot{δ}}_{v} \\ {\ddot{δ}}_{s} \\ {\ddot{δ}}_{t} \end{matrix}} + [\begin{matrix} C_{v v} & C_{v s} & 0 \\ C_{s v} & C_{s s} & C_{s t} \\ 0 & C_{t s} & C_{t t} \end{matrix}] {\begin{matrix} {\dot{δ}}_{v} \\ {\dot{δ}}_{s} \\ {\dot{δ}}_{t} \end{matrix}} + [\begin{matrix} K_{v v} & K_{v s} & 0 \\ K_{s v} & K_{s s} & K_{s t} \\ 0 & K_{t s} & K_{t t} \end{matrix}] {\begin{matrix} δ_{v} \\ δ_{s} \\ δ_{t} \end{matrix}} = {\begin{matrix} P_{v} \\ P_{s} \\ P_{t} \end{matrix}}

(10)

where [M], [C], and [K] are mass, damping, and stiffness matrix of the system;

{δ}

{\dot{δ}}

, and

{\ddot{δ}}

are the displacement, velocity, and acceleration vector; {p} is the load vector; and the subscripts v, s, and t represent the vehicle, SSFST, and tunnel, respectively.

The governing equations are solved by using the Newmark-β method of the stepwise integral method.²²

Experimental validation

A SSFST section of metro line 6 in Chongqing city, China, is selected. The arrangement of the measuring points is shown in Figure 11. The natural frequency of SSFST, structural displacements, and accelerations is analyzed. To simulate the field situation as much as possible, in the numerical models, the cross section of the tunnel is rectangular, and the stiffness values of the vibration isolators in the midspan and end of the slab are 6.66 kN/mm and 5.33 kN/mm, respectively. The measuring points of displacement and acceleration, compared to the points in the simulation model, are in the same position.

Figure 11.

Layout of the measuring points.

Modal validation

Natural frequency is an inherent property of the structure, which is related to the stiffness and mass of the structure. For the finite element model of SSFST, the modal analysis was carried out. The first 10 eigenmodes and natural frequency of SSFST are shown in Figure 12. It can be seen that the main function of SSFST is to provide low-frequency vibration isolation. The fundamental system frequency of SSFST lies in the range of 8–25 Hz.

Figure 12.

First 10 eigenmodes and natural frequency of steel-spring floating slab track.

The residual vibration method is conducted to measure the vertical natural frequency of SSFST, which is 7.37 Hz. It is close to the first-order modal frequency of 8.13 Hz obtained by modal analysis, and the mode shape is both a vertical mode. The results show that the stiffness and mass of the SSFST model are close to those of the field SSFST structure. It indicates that the model is reliable and can be used for subsequent dynamic analysis.

Comparison of dynamic results

Multiple field tests were conducted. The dynamic responses of the measuring points when 15 subway trains passed are selected as the analysis basis. The measured speed of subway trains is 87.9–92.5 km/h, and the average speed is 90 km/h.

Figure 13 shows the comparison between measured and simulated results of the displacement. Compared with the measured data, the displacement of slab and rail decreases by 7.8% and 9.9% under the single-carriage model and increases by 2.0% and 3.8% under the two-carriage model. It indicates that the displacement of the two-carriage model is much closer to the measured data and slightly larger than the measured displacement, which is relatively conservative and retains a certain safety margin for field operation.

Figure 13.

Comparison between measured and simulated results of displacement.

The root-mean-square (RMS) accelerations of the structures are compared. Figure 14 shows the comparison between the measured and simulated accelerations. In terms of the acceleration of rail, slab, or tunnel, the results under the two-carriage model are closer to the measured data. Especially for the rail acceleration, the rail acceleration from the single-carriage model is about 10 m/s², significantly smaller than the results of the two-carriage model and measured data, which are about 25 m/s² and 27.5 m/s², respectively. It illustrates that the coupling effect of loads from the front and rear bogies of adjacent carriages is obvious and cannot be ignored.

Figure 14.

Comparison between measured and simulated results of acceleration.

In general, the displacement and acceleration of the two-carriage model are closer to the dynamic response of the in situ SSFST, indicating that the two-carriage model has advantages in the simulation application.

Dynamic results analysis

Considering that the simulation of a whole vehicle will greatly increase the calculation time cost, only the dynamic response and vibration characteristics of the SSFST of one, two, and three carriages are analyzed.

Analysis of the dynamic deformation.

The displacements of the rail and slab at different positions are shown in Figure 15. In Figure 15(a) and (b), compared with under the single-carriage model, the maximum displacements of the rail and slab at the midspan increase to a certain extent under the multi-carriage model (two or three carriages). The maximum displacements of the rail and slab increase by 0.48 mm and 0.34 mm, respectively, due to the superimposition of multi-carriage dynamic loads. While the maximum displacement of the rail and slab at the end of the slab barely changes under the single-carriage and multi-carriage models, as shown in Figure 15(c) and (d). This is because of the failure of superimposition of multi-carriage dynamic loads due to the discontinuity between the slabs.

Figure 15.

Displacement of the rail and steel-spring floating slab track on different positions: (a) rail displacement at midspan; (b) slab displacement at midspan; (c) rail displacement at slab end; (d) slab displacement at slab end.

In addition, the maximum displacements of the rail and slab under two-carriage and three-carriage models are basically the same. Since there are at most two carriages on the same slab at the same time, the vehicle loads include no more than two carriages.

Analysis of vibration characteristics

Time domain analysis

The RMS accelerations at different positions under multi-carriages are shown in Figure 16(a). The accelerations of the concrete bed and tunnel increase slightly, and the accelerations of the rail and SSFST increase significantly with the increase of the carriage number. Figure 16(b) shows the accelerations of the rail and slab in time domain. The rail acceleration reaches 100 m/s² at the junction of the two carriages under the multi-carriage model. Affected by the vibration transfer caused by the load of the front carriage, the rail acceleration at the middle part of the second carriage reaches 50 m/s², larger than the acceleration of 25 m/s² at the middle part of the front carriage. Besides, the acceleration of the slab under the multi-carriage model is also slightly higher than that under the single-carriage model. In conclusion, the accelerations of rail and SSFST under the multi-carriage model are significantly different from those under the single-carriage model, and the difference cannot be ignored.

Figure 16.

Acceleration at different positions: (a) the RMS acceleration at different positions under different amounts of carriages; (b) acceleration in time domain: (i) rail acceleration; (ii) slab acceleration.

Frequency domain analysis

The vertical acceleration Z vibration level (VL_Z) of the SSFST is used as the evaluation indicator in this study, referring to the standard IS26311.²³ And VL_Z is expressed as

V L_{z} = 20 lg \frac{a_{r e f}}{a_{0}}

(11)

where

a_{r e f}

is the weighted RMS acceleration; a₀ is the reference acceleration, equal to 10⁻⁶ m/s².

The band of 4–250 Hz, the sensitive frequency scopes of the track system,²⁴ is selected, and the curve of VLz is obtained, as shown in Figure 17. VLz increases in the range of 32–250 Hz with the increase of the carriage number and reaches the maximum at around 40 Hz, which illustrates that the number of carriages will certainly increase the vibration of the SSFST.

Figure 17.

Vertical acceleration Z vibration leve: (a) VL_Z at midspan; (b) VL_Z at end of the slab.

The maximum weighted Z vibration level (VL_Zmax) and vibration reduction effectiveness (ΔVL_Z) under a different number of carriages are shown in Table 4. VL_Zmax increases with the increase of carriages. Compared with the results of the single-carriage model, ΔVL_Z increases by 1.4–2.0 dB under the multi-carriage model. However, ΔVL_Z under two-carriage and three-carriage models is consistent, which illustrates that considering more than two carriages does not affect the evaluation of vibration reduction.

Table 4.

Maximum weighted Z vibration level and vibration reduction effectiveness (unit: dB).

Position	Type	Single-carriage		Two-carriage		Three-carriage
Position	Type	VL_Zmax	ΔVL_Z	VL_Zmax	ΔVL_Z	VL_Zmax	ΔVL_Z
Slab end	Slab track	65.8	13.4	68.7	14.9	69.8	14.9
Slab end	SSFST	52.4	13.4	53.8	14.9	54.9	14.9
Midspan	Slab track	65.6	11.4	69.1	13.1	70.3	13.4
Midspan	SSFST	54.2	11.4	56.0	13.1	56.9	13.4

Note: SSFST: steel-spring floating slab track.

Conclusions

Based on the FEM and multi-body dynamics, a refined three-dimensional coupled model of multi-carriage-SSFST-tunnel is established. The differences in dynamic response and vibration characteristics of the SSFST between the single-carriage and multi-carriage models are analyzed. The conclusions are as follows:

Due to the low stiffness of SSFST, the coupling effect of the front and rear bogies of adjacent carriages cannot be ignored.

Compared with the single-carriage model, structural displacements and accelerations under the two-carriage model are closer to the measured data, which illustrate that the two-carriage model is closer to the actual situation and can more effectively guide the design and optimal layout of the SSFST.

Compared with the single-carriage model, the maximum displacements of the rail and slab in the midspan under the multi-carriage model increase by 0.48 mm and 0.34 mm, respectively, while the displacement in the end of the slab remains unchanged.

The VLz increases in the range of 32–250 Hz, and the ΔVL_Z increases by 1.4–2.0 dB under the multi-carriage model, which illustrates the vibration response, and vibration reduction effectiveness will increase if using the multi-carriage model.

In the multi-carriage model, there are no significant differences in structural deformation and dynamic response between two-carriage and three-carriage models. The maximum displacements of the rail and slab and the vibration reduction effectiveness remain unchanged.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by China National Key R&D Plan (2016YFB1200100), National Natural Science Foundation of China (51778050) and China Railway R&D Project (P2018G003).

ORCID iD

Wenhao Chang

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